3.1194 \(\int \frac{(a-i a x)^{7/4}}{(a+i a x)^{7/4}} \, dx\)

Optimal. Leaf size=291 \[ \frac{4 i (a-i a x)^{7/4}}{3 a (a+i a x)^{3/4}}+\frac{7 i \sqrt [4]{a+i a x} (a-i a x)^{3/4}}{3 a}-\frac{7 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}+\frac{7 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}+\frac{7 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}-\frac{7 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}} \]

[Out]

(((4*I)/3)*(a - I*a*x)^(7/4))/(a*(a + I*a*x)^(3/4)) + (((7*I)/3)*(a - I*a*x)^(3/
4)*(a + I*a*x)^(1/4))/a + ((7*I)*ArcTan[1 - (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a
*x)^(1/4)])/Sqrt[2] - ((7*I)*ArcTan[1 + (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^
(1/4)])/Sqrt[2] - (((7*I)/2)*Log[1 + Sqrt[a - I*a*x]/Sqrt[a + I*a*x] - (Sqrt[2]*
(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2] + (((7*I)/2)*Log[1 + Sqrt[a - I*a
*x]/Sqrt[a + I*a*x] + (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2]

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Rubi [A]  time = 0.315044, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36 \[ \frac{4 i (a-i a x)^{7/4}}{3 a (a+i a x)^{3/4}}+\frac{7 i \sqrt [4]{a+i a x} (a-i a x)^{3/4}}{3 a}-\frac{7 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}+\frac{7 i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{2 \sqrt{2}}+\frac{7 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}}-\frac{7 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(a - I*a*x)^(7/4)/(a + I*a*x)^(7/4),x]

[Out]

(((4*I)/3)*(a - I*a*x)^(7/4))/(a*(a + I*a*x)^(3/4)) + (((7*I)/3)*(a - I*a*x)^(3/
4)*(a + I*a*x)^(1/4))/a + ((7*I)*ArcTan[1 - (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a
*x)^(1/4)])/Sqrt[2] - ((7*I)*ArcTan[1 + (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^
(1/4)])/Sqrt[2] - (((7*I)/2)*Log[1 + Sqrt[a - I*a*x]/Sqrt[a + I*a*x] - (Sqrt[2]*
(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2] + (((7*I)/2)*Log[1 + Sqrt[a - I*a
*x]/Sqrt[a + I*a*x] + (Sqrt[2]*(a - I*a*x)^(1/4))/(a + I*a*x)^(1/4)])/Sqrt[2]

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Rubi in Sympy [A]  time = 48.4153, size = 250, normalized size = 0.86 \[ - \frac{7 \sqrt{2} i \log{\left (- \frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + \frac{\sqrt{- i a x + a}}{\sqrt{i a x + a}} + 1 \right )}}{4} + \frac{7 \sqrt{2} i \log{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + \frac{\sqrt{- i a x + a}}{\sqrt{i a x + a}} + 1 \right )}}{4} - \frac{7 \sqrt{2} i \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} - 1 \right )}}{2} - \frac{7 \sqrt{2} i \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- i a x + a}}{\sqrt [4]{i a x + a}} + 1 \right )}}{2} + \frac{4 i \left (- i a x + a\right )^{\frac{7}{4}}}{3 a \left (i a x + a\right )^{\frac{3}{4}}} + \frac{7 i \left (- i a x + a\right )^{\frac{3}{4}} \sqrt [4]{i a x + a}}{3 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a-I*a*x)**(7/4)/(a+I*a*x)**(7/4),x)

[Out]

-7*sqrt(2)*I*log(-sqrt(2)*(-I*a*x + a)**(1/4)/(I*a*x + a)**(1/4) + sqrt(-I*a*x +
a)/sqrt(I*a*x + a) + 1)/4 + 7*sqrt(2)*I*log(sqrt(2)*(-I*a*x + a)**(1/4)/(I*a*x
+ a)**(1/4) + sqrt(-I*a*x + a)/sqrt(I*a*x + a) + 1)/4 - 7*sqrt(2)*I*atan(sqrt(2)
*(-I*a*x + a)**(1/4)/(I*a*x + a)**(1/4) - 1)/2 - 7*sqrt(2)*I*atan(sqrt(2)*(-I*a*
x + a)**(1/4)/(I*a*x + a)**(1/4) + 1)/2 + 4*I*(-I*a*x + a)**(7/4)/(3*a*(I*a*x +
a)**(3/4)) + 7*I*(-I*a*x + a)**(3/4)*(I*a*x + a)**(1/4)/(3*a)

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Mathematica [C]  time = 0.0689304, size = 76, normalized size = 0.26 \[ \frac{(a-i a x)^{3/4} \left (-7 i \sqrt [4]{2} (1+i x)^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2}-\frac{i x}{2}\right )-3 x+11 i\right )}{3 (a+i a x)^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - I*a*x)^(7/4)/(a + I*a*x)^(7/4),x]

[Out]

((a - I*a*x)^(3/4)*(11*I - 3*x - (7*I)*2^(1/4)*(1 + I*x)^(3/4)*Hypergeometric2F1
[3/4, 3/4, 7/4, 1/2 - (I/2)*x]))/(3*(a + I*a*x)^(3/4))

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{1 \left ( a-iax \right ) ^{{\frac{7}{4}}} \left ( a+iax \right ) ^{-{\frac{7}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a-I*a*x)^(7/4)/(a+I*a*x)^(7/4),x)

[Out]

int((a-I*a*x)^(7/4)/(a+I*a*x)^(7/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-i \, a x + a\right )}^{\frac{7}{4}}}{{\left (i \, a x + a\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(7/4)/(I*a*x + a)^(7/4),x, algorithm="maxima")

[Out]

integrate((-I*a*x + a)^(7/4)/(I*a*x + a)^(7/4), x)

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Fricas [A]  time = 0.233022, size = 370, normalized size = 1.27 \[ \frac{6 i \, a x^{2} - 3 \, \sqrt{49 i}{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}} \log \left (\frac{\sqrt{49 i}{\left (a x + i \, a\right )} + 7 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{7 \, x + 7 i}\right ) + 3 \, \sqrt{49 i}{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}} \log \left (-\frac{\sqrt{49 i}{\left (a x + i \, a\right )} - 7 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{7 \, x + 7 i}\right ) - 3 \, \sqrt{-49 i}{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}} \log \left (\frac{\sqrt{-49 i}{\left (a x + i \, a\right )} + 7 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{7 \, x + 7 i}\right ) + 3 \, \sqrt{-49 i}{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}} \log \left (-\frac{\sqrt{-49 i}{\left (a x + i \, a\right )} - 7 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{7 \, x + 7 i}\right ) + 16 \, a x + 22 i \, a}{6 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(7/4)/(I*a*x + a)^(7/4),x, algorithm="fricas")

[Out]

1/6*(6*I*a*x^2 - 3*sqrt(49*I)*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)*log((sqrt(49*
I)*(a*x + I*a) + 7*(I*a*x + a)^(1/4)*(-I*a*x + a)^(3/4))/(7*x + 7*I)) + 3*sqrt(4
9*I)*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)*log(-(sqrt(49*I)*(a*x + I*a) - 7*(I*a*
x + a)^(1/4)*(-I*a*x + a)^(3/4))/(7*x + 7*I)) - 3*sqrt(-49*I)*(I*a*x + a)^(3/4)*
(-I*a*x + a)^(1/4)*log((sqrt(-49*I)*(a*x + I*a) + 7*(I*a*x + a)^(1/4)*(-I*a*x +
a)^(3/4))/(7*x + 7*I)) + 3*sqrt(-49*I)*(I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4)*log(
-(sqrt(-49*I)*(a*x + I*a) - 7*(I*a*x + a)^(1/4)*(-I*a*x + a)^(3/4))/(7*x + 7*I))
+ 16*a*x + 22*I*a)/((I*a*x + a)^(3/4)*(-I*a*x + a)^(1/4))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a-I*a*x)**(7/4)/(a+I*a*x)**(7/4),x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-I*a*x + a)^(7/4)/(I*a*x + a)^(7/4),x, algorithm="giac")

[Out]

Exception raised: TypeError